Time constant for heat transfer

It is frequently required to examine the combined performance of two or more processes in series, e.g. two systems or capacities, each described by a transfer function in the form of equations 7.19 or 7.26. Such multicapacity processes do not necessarily have to consist of more than one physical unit. Examples of the latter are a protected thermocouple junction where the time constant for heat transfer across the sheath material surrounding the junction is significant, or a distillation column in which each tray can be assumed to act as a separate capacity with respect to liquid flow and thermal energy. 

Additionally, we assume that the time constants for heat transfer and mass transport are of the same order of magnitude, i.e.,... 

The reactor is rather well behaved when the outside waU temperature is 335 K, but thermal runaway occurs when Twau = 336 K. Hence, Twaii exhibits a critical value between 335 and 336 K because thermal runaway occurs when Twaii > (rwaii)criticai- Thermal runaway can be prevented when Twau = 340 K if the surface-to-volume ratio of the reactor is increased by decreasing the tube radius R. This important design modification is accounted for by decreasing the time constant for heat transfer across the lateral surface. Numerical results are summarized in Table 4-2 for a single-pipe reactor with Tuaet = Rwaii = 340 K at nine different values of the heat transfer time constant. 

Hence, the time constant for heat transfer across the inner wall at radius inside is the same for the reactive fiuid in the inner tube (subscript Rx ) and the reactive cooling fluid in the outer tube (subscript cool ) ... 

The time constant for heat transfer to the surronndings, across the outer wall, is... 

The time constants characterizing heat transfer in convection or radiation dominated rotary kilns are readily developed using less general heat-transfer models than that presented herein. These time constants define simple scaling laws which can be used to estimate the effects of fill fraction, kiln diameter, moisture, and rotation rate on the temperatures of the soHds. Criteria can also be estabHshed for estimating the relative importance of radiation and convection. In the following analysis, the kiln wall temperature, and the kiln gas temperature, T, are considered constant. Separate analyses are conducted for dry and wet conditions. 

From a comparison of the times for heat transfer and heat production, it is possible to say that heat transfer will not be a problem at this scale, and temperature gradients should not be present since the liquid circulation time is relatively small compared to the time constant for heat production. 

Time constants. Where there is a capacity and a throughput, the measurement device will exhibit a time constant. For example, any temperature measurement device has a thermal capacity (mass times heat capacity) and a heat flow term (heat transfer coefficient and area). Both the temperature measurement device and its associated thermowell will exhibit behavior typical of time constants. 

In the case of a temperature probe, the capacity is a heat capacity C == me, where m is the mass and c the material heat capacity, and the resistance is a thermal resistance R = l/(hA), where h is the heat transfer coefficient and A is the sensor surface area. Thus the time constant of a temperature probe is T = mc/ hA). Note that the time constant depends not only on the probe, but also on the environment in which the probe is located. According to the same principle, the time constant, for example, of the flow cell of a gas analyzer is r = Vwhere V is the volume of the cell and the sample flow rate. 

We follow the analysis of Frank-Kamenetskii [3] of a slab of half-thickness, rG, heated by convection with a constant convective heat transfer coefficient, h, from an ambient of Too. The initial temperature is 7j < 7 ,XJ however, we consider no solution over time. We only examine the steady state solution, and look for conditions where it is not valid. If we return to the analysis for autoignition, under a uniform temperature state (see the Semenov model in Section 4.3) we saw that a critical state exists that was just on the fringe of valid steady solutions. Physically, this means that as the self-heating proceeds, there is a state of relatively low temperature where a steady condition is sustained. This is like the warm bag of mulch where the interior is a slightly higher temperature than the ambient. The exothermiscity is exactly balanced by the heat conducted away from the interior. However, under some critical condition of size (rG) or ambient heating (h and Too), we might leave the content world of steady state and a dynamic condition will... 

Hatim (H2) obtained numerical solutions for heat transfer from a sphere of constant temperature accelerating from rest. The trajectory was calculated from Eq. (11-33), and the time-dependent Navier-Stokes and energy equations... 

We take the heat transfer coefficient a to be independent of the jet velocity and of the residence time in the vessel. Physically, this assumption together with the assumption of complete mixing of the substance in the reaction vessel and of a constant mean temperature throughout the vessel corresponds to the idea that for heat transfer the governing factor is the thermal resistance from the internal wall of the vessel to the outside space in which the temperature is kept at T0. In other words, our assumption corresponds to the concept of a vessel which is thermally insulated from outside. 

By integrating Eq. 2, it is possible to derive the drying time in the constant rate period. Equation 2 is derived for heat transfer to the material being dried by circulating air. When large metal sheets or trays are close to the product, it is not possible to ignore the conduction and radiation contribution to heat transfer. In this case, the solid temperature is raised above the air wet-bulb temperature and Eq. 2 becomes ... 

The analytical solutions for transient conduction in plates, cylinders, and spheres have been obtained by Heisler [9] and the solutions represented graphically for more convenient use. These solutions are for the case of a solid of initially uniform temperature Tg exposed at time zero to a surrounding fluid medium at a constant temperature T. The surface of the solid is cooled or heated by the fluid with a constant convective heat-transfer coefficient h. The sohd is assumed to have a constant thermal conductivity and a constant thermal diffusivity a, defined as... 

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